Quiz

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Problems that look complicated can often be broken down into simpler scenarios that contain more information. Since the individual case is easier to consider, the whole problem becomes easier to solve.

Some questions may provide you with the different cases. Then your job will be to determine if each scenario is a solution or not. Other questions will require you to think of all possible scenarios. Then, keep in mind all the givens and all the restrictions, but most importantly, when listing all the possibilities, be systematic!

How many triples of positive integers are there such that \( a \times b + c = 8 \)?

If \( a = 1 \), then we can have \( b = 1, 2 \ldots 7,\) which gives us 7 solutions.
If \( a = 2 \), then we can have \( b = 1, 2,\) and \(3,\) which gives us 3 solutions.
If \( a = 3 \), then we can have \(b =1\) and \(2,\) which gives us 2 solutions.
If \( a =4, 5, 6\) or \(7,\) we can only have \(b = 1 \). This gives us 4 solutions.